11. The system described by the equation y(n)=ay(n+1)+b x(n) is a recursive system.
A. True
B. False
Answer: B
Since the present output depends on the value of the future output, the system is not called a Recursive system.
12. If the system is initially relaxed at time n=0 and memory equals zero, then the response of such state is called ____________
A. Zero-state response
B. Zero-input response
C. Zero-condition response
D. None of the mentioned
Answer: A
The memory of the system, describes, in some cases, the ‘state’ of the system, the output of the system is called as ‘zero-state response’.
13. Zero-state response is also known as ____________
A. Free response
B. Forced response
C. Natural response
D. None of the mentioned
Answer: B
The zero-state response depends on the nature of the system and the input signal. Since this output is a response forced upon it by the input signal, it is also known as a ‘Forced response’.
14. Zero-input response is also known as Natural or Free response.
A. True
B. False
Answer: A
For a zero-input response, the input is zero and the output of the system is independent of the input of the system. So, the response of such a system is also known as a Natural or Free response.
15. The solution obtained by assuming the input x(n) of the system is zero is ____________
A. General solution
B. Particular solution
C. Complete solution
D. Homogenous solution
Answer: D
By making the input x(n)=0 we will get a homogeneous difference equation and the solution of that differential equation is known as a Homogenous or Complementary solution.
16. What is the homogenous solution of the system described by the first-order difference equation y(n)+ay(n-1)=x(n)?
A. c(A.n(where ‘c’ is a constant)
B. c(A.-n
C. c(-A.n
D. c(-A.-n
Answer: C
The assumed solution obtained by assigning x(n)=0 is
yh(n)=λn
=>y(n)+ay(n-1)=0
=>λn+a λn-1=0
=>λn-1(λ+A.=0
=>λ=-a
=>yh(n)=cλn=c(-A.n
17. What is the zero-input response of the system described by the homogenous second order equation y(n)-3y(n-1)-4y(n-2)=0 if the initial conditions are y(-1)=5 and y(-2)=0?
A. (-1)n-1 + (4)n-2
B. (-1)n+1 + (4)n+2
C. (-1)n+1 + (4)n-2
D. None of the mentioned
Answer: B
Given difference equation is y(n)-3y(n-1)-4y(n-2)=0—-(1)
Let y(n)=λn
Substituting y(n) in the given equation
=> λn – 3λn-1 – 4λn-2 = 0
=> λn-2(λ2 – 3λ – 4) = 0
the roots of the above equation are λ=-1,4
Therefore, the general form of the solution of the homogenous equation is
yh(n)=C1 λ1n+C2 λ2n
=C1(-1)n+C2(4)n—-(2)
The zero-input response of the system can be calculated from the homogenous solution by evaluating the constants in the above equation, given the initial conditions y(-1) and y(-2).
From the given equation (1)
y(0)=3y(-1)+4y(-2)
y(1)=3y(0)+4y(-1)
=3[3y(-1)+4y(-2)]+4y(-1)
=13y(-1)+12y(-2)
From the equation (2)
y(0)=C1+C2 and
y(1)=C1(-1)+C2(4)=-C1+4C2
By equating these two set of relations, we have
C1+C2=3y(-1)+4y(-2)=15
-C1+4C2=13y(-1)+12y(-2)=65
On solving the above two equations we get C1=-1 and C2=16
Therefore the zero-input response is Yzi(n) = (-1)n+1 + (4)n+2.
18. What is the particular solution of the first order difference equation y(n)+ay(n-1)=x(n) where |a|<1, when the input of the system x(n)=u(n)?
A. \(\frac{1}{1+a}\) u(n)
B. \(\frac{1}{1-a}\) u(n)
C. \(\frac{1}{1+a}\)
D. \(\frac{1}{1-a}\)
Answer: A
The assumed solution of the difference equation to the forcing equation x(n), called the particular solution of the difference equation is
yp(n)=Kx(n)=Ku(n) (where K is a scale factor)
Substitute the above equation in the given equation
=>Ku(n)+aKu(n-1)=u(n)
To determine K we must evaluate the above equation for any n>=1, so that no term vanishes.
=> K+aK=1
=>K=\(\frac{1}{1+a}\)
Therefore the particular solution is yp(n)=\(\frac{1}{1+a}\) u(n).
19. What is the particular solution of the difference equation y(n)=\(\frac{5}{6}y(n-1)-\frac{1}{6}\)y(n-2)+x(n) when the forcing function x(n)=2n, n≥0 and zero elsewhere?
A. \(\frac{1}{5}\) 2n
B. \(\frac{5}{8}\) 2n
C. \(\frac{8}{5}\) 2n
D. \(\frac{5}{8}\) 2-n
Answer: C
The assumed solution of the difference equation to the forcing equation x(n), called the particular solution of the difference equation is
yp(n)=Kx(n)=K2nu(n) (where K is a scale factor)
Upon substituting yp(n) into the difference equation, we obtain
To determine K we must evaluate the above equation for any n>=2, so that no term vanishes.
=> 4K=\(\frac{5}{6}\)(2K)-\(\frac{1}{6}\) (K)+4
=> K=\(\frac{8}{5}\)
=> yp(n)=(8/5) 2n.
20. The total solution of the difference equation is given as _______________
A. yp(n)-yh(n)
B. yp(n)+yh(n)
C. yh(n)-yp(n)
D. None of the mentioned
Answer: B
The linearity property of the linear constant-coefficient difference equation allows us to add the homogeneous and particular solutions in order to obtain the total solution.